Nostr notes by Terence Tao

An example occured just today, in which we were treated to the ...

2026-09-03T18:53:18Z

In reply to nevent1q…rrp6
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An example occured just today, in which we were treated to the unedifying spectacle of no fewer than three separate AI companies race to announce their own improvement on the bounded gaps between prime result that I mentioned yesterday at https://mathstodon.xyz/@tao/117197525544971208 . These results are numerically stronger than Stadlmann's improvement, but I am very glad that Stadlmann was able to complete her analysis just in time before the problem became contaminated, as the insights that are easily extractable from her paper - in particular, the value of developing equidistribution estimates for numbers with a large smooth factor, as well as reinforcing the use of previous innovations such as the Polymath8b "epsilon enlargement trick" and the Baker-Harman introduction of the Harman sieve, would have been significantly harder to extract in the counterfactual situation in which the AI-generated proofs were released first (without any professionally written writeup to improve the signal-to-noise ratio), and Stadlmann's effort was subsequently abandoned. (3/3)

But the currently fashionable practice of pointing a powerful AI ...

2026-09-03T18:53:06Z

In reply to nevent1q…c3sk
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But the currently fashionable practice of pointing a powerful AI tool at the task of answering a problem X, unguided by any human expert in the field X resides in, has created an unprecedented divergence between the production of answers, and the production of insight, to the point where the two questions have become _negatively correlated_: further optimization in the direction of generating answers for X can now _decrease_ the amount of insight one could have gained from the study of X, thanks to factors discussed in previous posts, such as contamination of reasoning with the solution, degradation of the signal-to-noise ratio, or in reduction of incentives to analyze the problem further.

A priori, there is no reason that AI use _necessarily_ has to perform this tradeoff. For instance, the use of AI for literature search is already well established as a basic way to gain initial insight into a problem, or to connect any newly discovered technique to the literature. But the current incentive to race for priority for solutions above all other concerns works against such more deliberate use cases. (2/3)

In basic research, such as pure mathematics, one might naively ...

2026-09-03T18:52:52Z

In basic research, such as pure mathematics, one might naively expect that the natural question to ask with regards to a given problem X in a field is

"What is the answer to X?".

But in many cases the more valuable question is

"What can be learned from studying X?"

The answer to X itself can of course be one of the things learned in this process of study; but one can learn far more useful information besides, such as

* What are the main difficulties to overcome to resolve X?
* What new techniques can one discover in order to solve X?
* Why are existing techniques insufficient to solve the problem by itself?
* How does X relate to results in prior literature?
* Can one uncover new connections between X and other topics Y, Z, ...?
* What are some natural related or followup questions X', X'', ... to study?

Nevertheless, until recently the two questions were closely aligned, to the point where it was not really necessary to distinguish the two: the only practical route to solving a difficult problem was to first address many of the subquestions listed above. (1/3)

Fundamentally, problems in pure mathematics, such as the global ...

2026-09-03T15:39:22Z

In reply to nevent1q…5hu4
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Fundamentally, problems in pure mathematics, such as the global regularity problem, serve a different purpose than immediately practical problems, such as that of finding a cure to a specific disease, or increasing the energy efficiency of some engine. In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tends to spur further development of the field through the efforts to solve such problems, and then to digest any partial or complete solutions that emerge for further insights. Prematurely solving the problem by purely AI-powered methods can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole. (6/6)